Enter a value, the mean, and the standard deviation. You'll get the z-score (how many standard deviations the value sits from the mean) plus its percentile in a normal distribution.
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A z-score restates a value as its distance from the mean, counted in standard deviations. That is what lets you compare a score of 130 on one test with 700 on another: both become a position on this one curve.
How the z-score calculator works
A z-score re-expresses a raw value on a universal scale: distance from the mean, measured in standard deviations. That's what makes it powerful: a 130 on an IQ test and a 700 on the SAT math section look nothing alike, but both are z ≈ +2, so they represent roughly the same rarity. The calculator subtracts the mean from your value, divides by the standard deviation, and then uses the standard normal distribution to translate that z-score into a percentile.
The formula
z = (x − μ) ÷ σ
Here x is your raw value, μ is the mean of the distribution, and σ is its standard deviation (which must be greater than zero, since a distribution with no spread has no meaningful z-scores).
Worked example
IQ scores are designed with a mean of 100 and a standard deviation of 15. For a score of 130:
z = (130 − 100) ÷ 15 = 30 ÷ 15 = +2
A z-score of +2 lands at about the 97.7th percentile; only around 2.3% of people score higher. For comparison, a score of 115 gives z = +1, which is about the 84.1st percentile.
The 68-95-99.7 rule
For anything roughly bell-shaped, z-scores map to rarity in a way worth memorizing: about 68% of values fall within z = ±1, about 95% within ±2, and about 99.7% within ±3. So a z-score beyond ±2 puts a value in the most unusual 5% of the distribution, and beyond ±3 in the most unusual 0.3%. One caveat: this mapping assumes normality. Income, house prices, and social media follower counts are all heavily skewed, and for data like that a z-score of +3 can be far less rare than the rule suggests.
Frequently asked questions
What is a z-score?
A z-score expresses how far a value sits from the mean, measured in standard deviations. A z-score of 0 means the value equals the mean, +2 means two standard deviations above it, and −1.5 means one and a half below. It's a universal ruler that lets you compare values from completely different scales.
How do you calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation: z = (x - mean) / SD. The result counts how many standard deviations your value sits above the mean (positive) or below it (negative), which is what lets scores from completely different scales be compared on one ruler.
How do I convert a z-score to a percentile?
Look up the z-score in the standard normal distribution (this calculator does it for you). A z-score of 0 is the 50th percentile, +1 is about the 84th, +2 about the 98th, and −1 about the 16th. The percentile tells you what share of a normally distributed population falls below your value.
What is a good z-score?
It depends entirely on context: a high z-score is great for a test result and terrible for blood pressure. As a rule of thumb, values within ±1 are ordinary (about 68% of a normal population), beyond ±2 are notably unusual (outer 5%), and beyond ±3 are rare (outer 0.3%).
Can a z-score be negative?
Yes. A negative z-score simply means the value is below the mean. A z-score of −2 is exactly as far from the mean as +2, just on the low side. The sign tells you the direction; the size tells you the distance.
Does a z-score require a normal distribution?
Computing the z-score itself doesn't; it's just (value − mean) ÷ standard deviation, valid for any data. But converting it to a percentile does assume the data is roughly normal. If your distribution is heavily skewed, the percentile this calculator reports won't match your data's actual percentile.